Problem 248 - Precalculus

Question

For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval. $$ f(x)=x^{5}-2 x, \quad \text { between } x=1 \text { and } x=2 $$

Step-by-Step Solution

Answer

The polynomial has at least one zero in the interval (1, 2) as the sign changes from negative to positive.

Step 1: Evaluate the function at the endpoints

First, evaluate the polynomial function \( f(x) = x^5 - 2x \) at the endpoints of the given interval. Calculate \( f(1) \):\[f(1) = 1^5 - 2 \cdot 1 = 1 - 2 = -1\]Calculate \( f(2) \):\[f(2) = 2^5 - 2 \cdot 2 = 32 - 4 = 28\]

Step 2: Verify sign change in interval

Observe the sign of the function values we calculated:- At \( x = 1 \), \( f(1) = -1 \), which is negative.- At \( x = 2 \), \( f(2) = 28 \), which is positive.The function changes from negative to positive, indicating a sign change.

Step 3: Apply the Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function changes signs over an interval \([a, b]\), then there is at least one \( c \) in \([a, b]\) such that \( f(c) = 0 \).Since \( f(x) \) is a polynomial, it is continuous over all real numbers, including the interval \([1, 2]\). The sign change from \( -1 \) to \( 28 \) across this interval confirms that there is at least one zero of \( f(x) \) within the interval \( (1, 2) \).

Key Concepts

Polynomial Functions and Their Characteristics

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Most commonly, you will encounter polynomials that consist of a single variable, like the function in our example: \[f(x) = x^5 - 2x\] Here, the highest power of the variable, which is 5, defines this as a 5th-degree polynomial. Polynomial functions are important because of their smooth and continuous nature. They do not have breaks, jumps, or holes, regardless of the values the variable takes.Some key features of polynomial functions include:

• Continuity: They are always continuous across their domain. We'll discuss continuity in more detail in another section.
• Differentiability: Polynomials can be differentiated without limitations. This means you can find their slopes and tangents at any point on their graph.
• Smoothness: The graphs of polynomial functions are smooth curves. This smoothness makes them easy to visually analyze.

Understanding the Zero of a Function

The zero of a function, also known as a root, is a point where the function's value is zero. For a function \( f(x) \), any value 'c' that satisfies the equation \( f(c) = 0 \) is referred to as the function's zero. In simpler terms, it is where the graph of the function crosses the x-axis.Identifying zeros is crucial because they often represent solutions to equations and give insights into the behavior of functions. For polynomial functions, finding zeros can reveal important information about factors and intercepts of the function's graph.To determine whether a polynomial function has a zero within a specific interval, the Intermediate Value Theorem is commonly used. Given a continuous function and a sign change over the interval, there must be at least one zero in that interval. This is what we applied in the original exercise to confirm a zero in \([1, 2]\).

The Nature of Continuous Functions

A continuous function is one where small changes in the input produce small changes in the output. There are no abrupt changes or interruptions, making these functions predictable and easy to understand.Polynomial functions are a prime example of continuous functions. They are defined everywhere on the real number line and have no breaks or holes. This property is essential when applying the Intermediate Value Theorem since the theorem only holds for continuous functions.With the polynomial function \( f(x) = x^5 - 2x \), continuity is inherent due to its polynomial nature, meaning that it does not require any additional checking beyond its polynomial form to know it is continuous across any interval including \([1, 2]\).

Sign Change and Its Implications

A sign change in a function's value over an interval indicates the possible presence of a zero of the function in that interval. This is because a sign change means that the function's values move from negative to positive (or vice versa), crossing the x-axis at least once in the process.In exercise terms, with function \( f(x) = x^5 - 2x \), we calculated the values at the endpoints of the interval:

• At \( x = 1 \), \( f(1) = -1 \)
• At \( x = 2 \), \( f(2) = 28 \)

The shift from \(-1\) to \(28\) demonstrates a clear sign change from negative to positive. By the Intermediate Value Theorem, in any scenario where a continuous function changes sign, there is at least one point where the function's value is zero, thus confirming the existence of a zero within the interval \((1, 2)\). This principle is pivotal in understanding functional behavior over selected intervals.